Construction of a Class of Forward Performance Processes ...sircar.princeton.edu/.../2019/12/fpp_Nov2019_ · Processes in Stochastic Factor Models and an Extension of Widder’s Theorem

Construction of a Class of Forward PerformanceProcesses in Stochastic Factor Models and an Extension

of Widder’s Theorem

Levon Avanesyan Mykhaylo Shkolnikov1 Ronnie Sircar

Princeton University

Abstract

We consider the problem of optimal portfolio selection under forward investmentperformance criteria in an incomplete market. Given multiple traded assets, theprices of which depend on multiple observable stochastic factors, we construct alarge class of forward performance processes, as well as the corresponding optimalportfolios, with power-utility initial data and for stock-factor correlation matriceswith eigenvalue equality (EVE) structure, which we introduce here. This is doneby solving the associated non-linear parabolic partial differential equations (PDEs)posed in the “wrong” time direction. Along the way we establish on domains an ex-plicit form of the generalized Widder’s theorem of Nadtochiy and Tehranchi [NT15,Theorem 3.12] and rely hereby on the Laplace inversion in time of the solutions tosuitable linear parabolic PDEs posed in the “right” time direction.

1 Introduction

In this paper we study the optimal portfolio selection problem under forwardinvestment criteria of power-utility form in incomplete markets, specifically stochas-tic factor models with a stock-factor correlation structure named EVE, which weintroduce here. Our setup is that of a continuous-time market model with multiplestocks whose returns and volatilities are functions of multiple observable stochasticfactors following jointly a diffusion process. The incompleteness arises hereby fromthe imperfect correlation between the Brownian motions driving the stock pricesand the factors. The factors themselves can model various market inputs, includingstochastic interest rates, stochastic volatility and major macroeconomic indicators,such as inflation, GDP growth or the unemployment rate.

The optimal portfolio problem in continuous time was originally considered byMerton in his pioneering work [Mer69], [Mer71], and is commonly referred to asthe Merton problem. In this framework an investor looks to maximize her expectedterminal utility from wealth acquired in the investment process within a geometricBrownian motion market model. Good compilations of classical results can be found

2010 Mathematics Subject Classification. Primary: 35K55, 91G10; secondary: 35J15, 60H10.Key words and phrases. Factor models, forward performance processes, generalized Widder’s

theorem, Hamilton-Jacobi-Bellman equations, ill-posed partial differential equations, incompletemarkets, Merton problem, optimal portfolio selection, positive eigenfunctions, time-consistency.

1M. Shkolnikov was partially supported by the NSF grant DMS-1506290.

1

Forward Performance Processes

in the books [Duf10], [KS98]. As fundamental as this setup is, it has two importantdrawbacks. First, the investor must decide on her terminal utility function beforeentering the market, and thereby cannot adapt it to changes in market conditions.Second, before settling on an investment strategy, the investor must firmly set hertime horizon. That is, the portfolio derived in this framework is optimal only forone specific utility function over one time horizon.

External factors such as the economic cycle, natural disasters, and the politicalclimate can lead to dynamic changes in one’s preferences. This would change the ter-minal utility function, thereby affecting the optimal portfolio allocation. Moreover,the investor might want to alter the terminal time itself. In order to solve portfoliooptimization problems with an uncertain investment horizon, forward investmentperformance criteria were introduced and developed in [MZ06] and [MZ07], as wellas in [HH07]. Instead of looking to optimize the expectation of a deterministic util-ity function at a single terminal point in time, this approach looks to maximize theexpectation of a stochastic utility function at every single point in time. Forwardperformance processes (FPPs), as defined in [MZ10a], capture the time evolutionsof such stochastic utility functions.

A comprehensive description of all FPPs remains a challenging open problem.Much work towards this goal has been carried out throughout the last ten years, see[BRT09], [EKM13a], [EKM13b], [HH07], [MZ10c], and [Zit09] for some importantresults. In [MZ10c], Musiela and Zariphopoulou proposed a construction of FPPs bymeans of solutions to a stochastic partial differential equation (SPDE). To find allthe FPPs characterized by the SPDE, one would have to find all forward volatilityprocesses, along with initial utility functions, for which the SPDE has a classicalsolution. The case of zero forward volatility yields time-monotone FPPs, and wasextensively discussed in [MZ10a] and [MZ10b], as well as more recently in [KOZ18]in the presence of model uncertainty.

We consider factor-driven market models and FPPs into which the randomnessenters only through the underlying stochastic factors. Assuming such a form, witha compatible forward volatility process, the SPDE mentioned above reduces to anHJB equation set in the “wrong” time direction. In a complete market one canuse the Fenchel-Legendre transform to linearize the HJB equation, and arrive at alinear second-order parabolic PDE set in the “wrong” time direction (see [NT15]).In an incomplete market no such linearizing transformation is available in general.To the best of our knowledge, the only exception is the special case of power utilityin a one-factor market model, where a linearization is possible through a distor-tion transformation, as discovered in [Zar01] for the Merton problem, and used forthe construction of FPPs in [NT15], [NZ14], and [SSZ16]. Construction and rep-resentation of FPPs in multi-factor incomplete market models have recently beenaddressed in [SSZ16] and [LZ17]. The former deals with a two-factor case, and pro-vides asymptotic results for different time scales. The latter allows for an arbitrarynumber of factors and trading constraints, and gives backward stochastic differentialequation (BSDE) representations of FPPs. All of these papers assume power-type(or homothetic) utility structure, as we will do also in this paper.

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We introduce a new class of multi-factor market models, which we will call EVEcorrelation models (see Definition 2.4). In this framework we reduce the fully non-linear HJB equation to a linear second-order parabolic PDE. Thereby, we obtainexplicit characterizations of FPPs in such models. Our analysis also applies to theMerton problem, whose value function solves the same HJB equation posed in the“right” time direction.

In one-factor market models, Nadtochiy and Tehranchi [NT15, Theorem 3.12]exhibited a characterization of all positive solutions to the linear parabolic equationsposed in the “wrong” time direction, that arise in the construction of FPPs of power-utility type. Their theorem constitutes a generalization of the celebrated Widder’stheorem (see [Wid63]), which describes all positive solutions of the heat equationset in the “wrong” time direction. The generalized Widder’s theorem reveals thatpositive solutions of a linear second-order parabolic equation set in the “wrong” timedirection must be linear combinations of exponentially scaled positive eigenfunctionsfor the corresponding elliptic operator according to a positive finite Borel measure.Moreover, each solution is uniquely identified with a pairing of the eigenfunctionsand the measure.

In our first main theorem (Theorem 2.14) we give a new version of [NT15, The-orem 3.12] on domains in the multi-stock multi-factor EVE setup with an initialutility function of power type to describe a new class of FPPs. Note that gen-eralized Widder’s theorems do not provide a way to construct the pairings of theeigenfunctions and the measure. Our second set of results (see Theorem 2.17 andRemark 2.18) addresses this issue: in Theorem 2.17 we give the Laplace transformof the measure in terms of the solution to a linear parabolic equation set in the“right” time direction, and we provide a method (see Remark 2.18) of finding theonly possible corresponding eigenfunctions as well. Thus, we indeed obtain a largeexplicit class of FPPs.

The rest of the paper is structured as follows. In Section 2 we state our mainresults, postponing their proofs to later sections. In Section 3 we introduce relevantfacts about FPPs and subsequently prove Theorem 2.14. In Section 4 we showTheorem 2.17, summarize some results from the theory of linear elliptic operators,and use them to establish Propositions 2.21, 2.23, 2.26 and 2.27. In Section 5 wediscuss the Merton problem within the framework of our market model. Lastly, inSection 6 we discuss EVE correlation models and construct explicit FPPs in affinemulti-stock multi-factor market models of EVE type.

2 Main results

2.1 Model

Consider an investor with initial capital X0 = x > 0 aiming to invest in amarket with n ≥ 1 stocks, the prices of which follow a process S, and a risklessbank account with zero interest rate. The stock prices depend on an observablek-dimensional stochastic factor process Y taking values in D ⊆ Rk, and are driven

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by a dW -dimensional standard Brownian motion W . The factor process Y is itselfdriven by a dB-dimensional standard Brownian motion B, whose correlation with Wis given by a matrix corr(W,B) = (ρij)

dW ,dBi,j=1 := ρ, where ρij ∈ [−1, 1]. Without loss

of generality we assume that dW ≥ n (see [Kar97, Remark 0.2.6]). The investor’sfiltration (Ft)t≥0 is generated by a pair (S, Y ) of processes satisfying

dSitSit

= µi(Yt) dt+

dW∑j=1

σji(Yt) dW jt , i = 1, 2, . . . , n, (2.1)

dYt = α(Yt) dt+ κ(Yt)T dBt, (2.2)

Bt = ρTWt +ATW⊥t , (2.3)

where the superscript T denotes transposition and W⊥ is a dW⊥-dimensional stan-dard Brownian motion independent of W . We write µ for (µ1, µ2, . . . , µn)T and σ

for (σij)dW ,ni,j=1 throughout.

Remark 2.1. It is straightforward to show that the positive semidefiniteness of thecorrelation matrix of the Brownian motion (W, B) implies that the singular valuesof ρ are in [0, 1].

For the convenience of the reader we summarize the dimensions of all the quan-tities we have introduced thus far:

µ(·)− n× 1, σ(·)− dW × n, Wt − dW × 1, α(·)− k × 1, κ(·)− dB × k,Bt − dB × 1, ρ− dW × dB, A− dW⊥ × dB, W⊥t − dW⊥ × 1.

Note that there is no loss of generality in using the representation (2.3) for thestandard Brownian motion B, since we can let A be the square root of the positivesemidefinite matrix IdB − ρTρ (recall that the singular values of ρ belong to [0, 1]),and dW⊥ = dB.

Assumption 2.2. The functions µ : D → Rn, σ : D → RdW×n are continuous,and the stochastic differential equation (SDE) (2.2) possesses a unique weak solution.Moreover, the columns of ρ belong to the range of left-multiplication by σ(y) for ally ∈ D.

Remark 2.3. The last condition in Assumption 2.2 holds only if the column rankof ρ is less than or equal to the column rank of σ, and implies σ(y)σ(y)−1ρ = ρfor all y ∈ D, where σ(y)−1 is the Moore-Penrose pseudoinverse of σ(y). Indeed,σ(y)σ(y)−1σ(y) = σ(y), so that the columns of σ(y) (and consequently the vectors intheir span, that is, the range of the left-multiplication by σ(y)) are invariant underthe left-multiplication by σ(y)σ(y)−1.

Our main result is for a particular class of multi-factor models, which we definenext.

Definition 2.4. We will call a market model an eigenvalue equality (EVE) corre-lation model if for some p ∈ [0, 1],

ρTρ = p IdB . (2.4)

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Note that in Definition 2.4, p has to be between 0 and 1 since the singular valuesof ρ are in [0, 1] (see Remark 2.1).

Remark 2.5. The name EVE comes from the fact that the only restriction is onthe eigenvalues of the matrix ρTρ. For any orthonormal dB × dB matrix O, we mayreplace κ(·) by Oκ(·) and B by B = OB in (2.2) without changing the dynamicsof the pair (S, Y ). Since B is a dB-dimensional standard Brownian motion andcorr(W, B) = OTρTρO is diagonal for an appropriate choice of O, we could haveassumed without loss of generality from the very beginning that ρTρ is diagonal.

Remark 2.6. Note that for EVE correlation models, since ρ is a dW × dB-matrix,at least one of the following two has to hold:

(i) dW ≥ dB,

(ii) p = 0.

Finally, we remark that when dB = 1, ρ is a vector and p := ρTρ ∈ [0, 1], so that(2.4) holds automatically.

Section 6 is devoted to a further discussion of EVE correlation models.

2.2 Forward Performance Processes

The investor dynamically allocates her wealth in the market using a self-financingtrading strategy that at any time t ≥ 0 yields a portfolio allocation πt = (π1

t , . . . , πnt )

among the n stocks with the associated wealth process

dXπt

Xπt

= (σ(Yt)πt)Tλ(Yt) dt+ (σ(Yt)πt)

T dWt, Xπ0 = x, (2.5)

where λ(Yt) = (σ(Yt)T )−1µ(Yt) is the Sharpe ratio. Apart from the self-financeability,

we impose additional conditions on the trading strategies to ensure that their wealthprocesses Xπ are well-defined by (2.5).

Definition 2.7. An (Ft)t≥0-progressively measurable self-financing trading strategyis called admissible if its portfolio allocation π among the n stocks fulfills

∀ t ≥ 0 :

ˆ t

0

∣∣πTs σ(Ys)Tλ(Ys)

∣∣ds <∞ and

ˆ t

0

∣∣σ(Ys)πs∣∣2 ds <∞ (2.6)

with probability one. In this case, we write π ∈ A.

Next we recall the definition of FPPs given in [MZ10a]. These capture how theutility function of an investor evolves over time as she continues to invest in thefinancial market.

Definition 2.8. An (Ft)t≥0-progressively measurable process U·(·) : [0,∞)×(0,∞)→R is referred to as a (local) forward performance process (FPP) if

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(i) with probability one, all functions x 7→ Ut(x), t ≥ 0 are strictly concave andincreasing,

(ii) for each π ∈ A, the process Ut(Xπt ), t ≥ 0 is an (Ft)t≥0 (local) supermartingale,

(iii) there exists an optimal π∗ ∈ A for which Ut(Xπ∗t ), t ≥ 0 is an (Ft)t≥0 (local)

martingale.

We refer to [MZ10a], [MZ10c] and [NZ14] for motivation and explanation of thisdefinition. We consider (local) FPPs of factor-form into which the randomnessenters only through the stochastic factor process, that is,

Ut(x) = V (t, x, Yt), t ≥ 0 (2.7)

for a deterministic function V : [0,∞) × (0,∞) ×D → R. Throughout the paper,we look for FPPs where the initial utility function is of product form, and a powerfunction in the wealth variable:

U0(x) = V (0, x, Y0) = γγx1−γ

1− γh(Y0) for some γ ∈ (0,∞)\1. (2.8)

The crucial simplification arising from the structure in (2.8) lies in its propagationto positive times. In this paper we will construct (local) FPPs of the following form.

Definition 2.9. We will say a (local) FPP U·(·) is of separable power factor formif

Ut(x) = V (t, x, Yt) = γγx1−γ

1− γg(t, Yt), (2.9)

for some g that is continuously differentiable in t (its first argument) and twicecontinuously differentiable in y (the second argument).

In this paper, we characterize all separable power factor form local FPPs forEVE correlation models introduced in Definition 2.4.

2.3 Characterizing separable power factor form FPPs

In order to describe our construction of separable power factor form FPPs, weneed to introduce some quantities related to linear elliptic operators of the secondorder. Consider on C2(D) such an operator

L =1

2

k∑i,j=1

aij(y)∂yiyj +k∑i=1

bi(y)∂yi + P (y) (2.10)

under the following assumption.

Assumption 2.10. There exists a C3-diffeomorphism Ξ : D → Rk so that thefunctions

aij(z) :=((∇Ξi)

Ta∇Ξj)(Ξ−1(z)),

bi(z) :=((∇Ξi)

T b)(Ξ−1(z)) +

1

2trace

(Hess(Ξi) a

)(Ξ−1(z)),

P (z) := P (Ξ−1(z))

(2.11)

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are uniformly bounded and uniformly η-Holder continuous over Rk and the matricesa(z) := (aij(z))

ki,j=1 are non-degenerate uniformly in z ∈ Rk. That is, with the

notation b(·) = (b1(·), b2(·), . . . , bk(·))T and for some η ∈ (0, 1),

(i) supz∈Rk |a(z)|, supz∈Rk |b(z)|, supz∈Rk |P (z)| <∞,

(ii) ‖a‖η,Rk , ‖b‖η,Rk , ‖P‖η,Rk <∞, where ‖f‖η,Rk = supz,z′∈Rk, z 6=z′|f(z)−f(z′)||z−z′|η ,

and

(iii) infz∈Rk, |v|=1 vTa(z)v > 0.

Remark 2.11. Assumption 2.10 entails that the domain D is C3-diffeomorphic toRk and that the operator L on D can be obtained as a pushforward under a C3-diffeomorphism of a uniformly elliptic operator

L =1

2

k∑i,j=1

aij(z)∂zizj +

k∑i=1

bi(z)∂zi + P (z) (2.12)

on Rk with uniformly bounded and uniformly η-Holder continuous coefficients. For astar-shaped domain D, it is well-known (see, e.g., [Fer08, Subsection 10.1]) that onecan find C∞-diffeomorphisms Ξ−1 mapping Rk onto D. However, whether a locallyuniformly elliptic operator L with locally bounded and locally η-Holder continuouscoefficients is a pushforward under Ξ−1 of an operator L with coefficients satisfyingthe conditions (i)-(iii) of Assumption 2.10 needs to be checked on a case-by-casebasis. As an example, consider an operator 1

2

∑ki=1 y

cii (1 − yi)

c′i ∂yiyi + P (y) on(0, 1)k with constants ci, c

′i ∈ (4,∞) and a bounded η-Holder continuous potential P .

Then, for the C∞-diffeomorphism Ξ : (0, 1)k → Rk, y 7→ (tan(πyi − π/2))ki=1 it iselementary to verify that the coefficients of the resulting

L =1

2

k∑i=1

π2ycii (1− yi)c′i

cos(πyi − π/2)4

∣∣∣∣yi=arctan zi

∂zizi

+

k∑i=1

π2ycii (1− yi)c′i sin(πyi − π/2)

cos(πyi − π/2)3

∣∣∣∣yi=arctan zi

∂zi + P((arctan zi)

ki=1

) (2.13)

fulfill the conditions (i)-(iii) of Assumption 2.10.

Remark 2.12. Whenever D = Rk (as for instance in [NT15, Section 3.1]) it isstandard in the literature to assume that the conditions (i)-(iii) of Assumption 2.10hold for a(·), b(·), P (·). This implies the set of conditions in Assumption 2.10 bytaking Ξ to be the identity map. Moreover, in this case, the SDE (2.2) admits aunique weak solution (see [KS91, Chapter 5, Remarks 4.17 and 4.30]).

We define the Holder space C2,η(D) ⊂ C2(D) as the subspace consisting offunctions whose second-order partial derivatives are η-Holder continuous (in thesame sense as in condition (ii) of Assumption 2.10) on compact subsets of D. Next,

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we introduce the sets of positive eigenfunctions for the operator L, which correspondto eigenvalues ζ ∈ R, and are normalized at some fixed y0 ∈ D:

CL−ζ(D) =ψ ∈ C2,η(D) : ψ(·) > 0, ψ(y0) = 1, (L − ζ)ψ = 0

. (2.14)

Moreover, we let SL(D) be the spectrum of L associated with positive eigenfunctions:

SL(D) =ζ ∈ R : CL−ζ(D) 6= ∅

. (2.15)

In subsection 4.2, we provide some well-known results about the structure of theeigenfunction spaces CL−ζ(D) and the set of eigenvalues SL(D). In particular,Proposition 4.4 yields that in our setup SL(D) is a half-line.

Finally, we call a functional Ψ : SL(D)×D → (0,∞) such that Ψ(ζ, ·) ∈ CL−ζ(D)for all ζ ∈ SL(D), a selection of positive eigenfunctions, and recall the definition ofBochner integrability in this setting.

Definition 2.13. Given a positive finite Borel measure ν on SL(D), we refer to a se-lection of positive eigenfunctions Ψ : SL(D)×D → (0,∞) as ν-Bochner integrable if,for all compact K ⊂ D,

´SL(D) ‖Ψ(ζ, ·)‖K ν(dζ) <∞, where ‖f‖K = supy∈K |f(y)|.

In preparation for our main result we define

a(·) = κ(·)Tκ(·), b(·) = α(·) + Γκ(·)TρTλ(·), P (·) =Γ

2qλ(·)Tλ(·), (2.16)

where Γ = 1−γγ and q = 1

1+Γp .

Theorem 2.14. Consider an EVE correlation model (2.1)-(2.3) with a correlationmatrix ρ satisfying Assumption 2.2. Suppose the second-order linear elliptic operatorL in (2.10) with coefficients provided in (2.16) satisfies Assumption 2.10. Then,given a function h : D → (0,∞), there exists a local FPP of separable power factorform with the initial condition

U0(x) = γγx1−γ

1− γh(Y0)q (2.17)

if and only if there exists a positive finite Borel measure ν on SL(D) and a ν-Bochnerintegrable selection of positive eigenfunctions Ψ : SL(D)×D → (0,∞) such that

h(y) =

ˆSL(D)

Ψ(ζ, y) ν(dζ). (2.18)

Furthermore, each local FPP of separable power factor form is uniquely identified bysuch a pairing (Ψ, ν), and is given by

Ut(x) = γγx1−γ

1− γ

(ˆSL(D)

e−tζΨ(ζ, Yt) ν(dζ)

)q. (2.19)

Any π∗ that solves

σ(Yt)π∗t =

1

γ

(λ(Yt) + qρκ(Yt)

´SL(D) e

−tζ (∇yΨ)(ζ, Yt) ν(dζ)´SL(D) e

−tζ Ψ(ζ, Yt) ν(dζ)

)(2.20)

is an associated optimal portfolio.

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Remark 2.15. We note that the equation (2.20) for optimal portfolios π∗ does notinvolve the initial wealth x. This is a consequence of the local FPP being of separablepower factor form. In the setting of the Merton problem, the same statement is true(and well-known) for terminal utility functions of power form.

Remark 2.16. A solution to the optimal portfolio equation (2.20) can be obtainedas follows. Since σ(·)−1 = (σ(·)Tσ(·))−1σ(·)T , one can write λ(·) = (σ(·)T )−1µ(·) asσ(·)(σ(·)Tσ(·))−1µ(·). In addition, by Assumption 2.2 and the Borel selection resultof [Bog07, Theorem 6.9.6], one can find a measurable ς : D → Rn×dB satisfyingσ(·)ς(·) = ρ, which renders

π∗t =1

γ

((σ(Yt)

Tσ(Yt))−1µ(Yt) + qς(Yt)κ(Yt)

´SL(D) e

−tζ (∇yΨ)(ζ, Yt) ν(dζ)´SL(D) e

−tζ Ψ(ζ, Yt) ν(dζ)

)(2.21)

a solution of (2.20).

The above theorem shows that for given admissible initial conditions, one canconstruct separable factor-form FPPs in EVE correlation models with general factordomains D ⊆ Rk, while also providing necessary and sufficient conditions for suchadmissibility. In particular, an investor with risk-aversion γ and dependence h(Y0) ofher current utility function on the initial value of the factor process, can extrapolatethe future values of her utility function according to (2.19) and acquire a portfoliofulfilling (2.20) (e.g. the portfolio in (2.21)), provided h is of the form (2.18). It istherefore crucial to understand which functions h admit the representation (2.18)and to be able to determine the pairings (Ψ, ν) for such.

Note that condition (iii) in Assumption 2.10 and the invertibility of the Jacobianmatrix of Ξ yield that κ(y) has full column rank k at each point y, and thus k ≤ dB.If p 6= 0, this combined with the observations in Remarks 2.3 and 2.6, implies thedimensional relationship k ≤ dB ≤ n ≤ dW in our model (2.1)-(2.3).

To the best of our knowledge, the only other paper addressing explicitly FPPsin multi-factor models is [LZ17]. In the models they consider, the factors are expo-nentially ergodic and live on the full space D = Rk, and the following dimensionalrelationship holds: n ≤ dW = dB = k. In addition, the form of the SPDE in[LZ17] (compare [LZ17, equation (10)] to e.g. [NT15, equation (3)]) implies thatσσ−1 = IdW , and thereby n = dW . Moreover, in [LZ17], ρ = corr(W,B) = IdW ,and thus their model fits into the EVE framework with p = 1. The main differenceof the setup in [LZ17] from ours is the possibility of constraints on the set of ad-missible portfolios. Without constraints, it is possible to linearize the semi-linearPDE in [LZ17, equation (13)] through the exact same steps as in the proof of ourProposition 3.3 below. For general constraints this is not possible. The authors cir-cumvent this issue by representing FPPs as functions of the solutions to appropriateinfinite-horizon BSDEs instead.

Another major difference from our results is in the set of allowable initial condi-tions from which FPPs can be constructed. In the absence of constraints, the resultsin [LZ17] require the measure ν (in the terminology of our Theorem 2.14) to be a

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multiple of a Dirac mass on an element of the set of eigenvalues SL(Rk), therebyrestricting the function h to be a positive eigenfunction of the elliptic operator L.Our Theorem 2.14 characterizes all admissible initial conditions through the equa-tion (2.18). In addition, our factors live on general domains D ⊆ Rk and are notrequired to be ergodic.

2.4 Finding selections of positive eigenfunctions Ψ and measures ν

The next set of results addresses the problem of solving the equation (2.18) forthe pairing (Ψ, ν), when it exists. The equation (2.18) stems from a new variant ofthe generalized Widder’s theorem of Nadtochiy and Tehranchi [NT15, Theorem 3.12](see Theorem 3.4 below) and, thus, our results can be viewed as yielding explicitversions of such theorems. The following theorem is also of independent interest, asit relates the pairing (Ψ, ν) arising in the positive solution of a linear second-orderparabolic PDE posed in the “wrong” time direction to the solution of the same PDEposed in the “right” time direction.

Theorem 2.17. Let L satisfy Assumption 2.10 and let h ∈ C2,η(D) be a positivefunction such that

(t, y) 7→ E[h(Zt) 1τ>t

∣∣Z0 = y]

is locally bounded on [0, ε]×D (2.22)

for the weak solution Z of the SDE associated with L0 := L−P (y) and ε > 0, whereτ is the first exit time of Z from D. Then, there is a positive classical solution of

∂tu+ Lu = 0 pointwise on [−ε, 0]×D, with u(0, ·) = h. (2.23)

For a positive finite Borel measure ν on SL(D) and a ν-Bochner integrable selectionof positive eigenfunctions Ψ : SL(D)×D → (0,∞), the function h can be expressedas´SL(D) Ψ(ζ, ·) ν(dζ) if and only if the problem (2.23) has a unique positive classical

solution u, so that for every y ∈ D, the function u(·, y) on (−ε, 0] is the Laplacetransform of the measure Ψ(ζ, y) ν(dζ), that is,

u(t, y) =

ˆSL(D)

e−ζt Ψ(ζ, y) ν(dζ), t ∈ (−ε, 0]. (2.24)

In this case, it holds, in particular,

u(t, y0) =

ˆSL(D)

e−ζt ν(dζ), t ∈ (−ε, 0]. (2.25)

Remark 2.18. Theorem 2.17 reveals that, whenever a pairing (Ψ, ν) exists, it canbe inferred by finding the measure ν through a one-dimensional Laplace inversionof u(·, y0) (recall that the values of the Laplace transform on a non-trivial inter-val determine the underlying positive finite Borel measure, see e.g. [Bil12, Section30]) and then the functions Ψ(·, y), y ∈ D\y0 from u(·, y), y ∈ D\y0 throughadditional one-dimensional Laplace inversions.

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As a by-product we obtain the following uniqueness result for linear second-orderparabolic PDEs posed in the “wrong” time direction by combining the generalizedWidder’s theorem on domains (Theorem 3.4 below) with Theorem 2.17 and theuniqueness of the Laplace transform ([Bil12, Section 30]).

Corollary 2.19. For any operator L satisfying Assumption 2.10 and positive h ∈C2,η(D) such that the function in (2.22) is locally bounded on a non-trivial cylinder[0, ε]×D, there is at most one positive solution u of the problem

∂tu+ Lu = 0 on [0,∞)×D, u(0, ·) = h. (2.26)

Remark 2.20. We stress that Corollary 2.19 is not an immediate consequence ofthe generalized Widder’s theorem on domains (Theorem 3.4) by itself. The lat-ter does ensure that every pairing (Ψ, ν) corresponds to exactly one positive so-lution u of (2.26). However, it is not clear a priori whether the representationh =´SL(D) Ψ(ζ, ·) ν(dζ) is unique for all functions h with the property (2.22). The-

orem 2.17 and the uniqueness of the Laplace transform ([Bil12, Section 30]) showthat this representation is, indeed, unique.

For arbitrary operators relatively little is known about the sets of positive eigen-functions CL−ζ(D). Nevertheless, in certain situations additional information onthe sets CL−ζ(D) is available and can be exploited to find the selection of positiveeigenfunctions Ψ for a given function h by a finite number of Laplace inversions.

Proposition 2.21. Let L satisfy Assumption 2.10, then

ζc(D) := infζ ∈ R : ζ ∈ SL(D)

∈ SL(D). (2.27)

If, in addition, the potential P is constant and L0 := L − P is such that the corre-sponding solution of the generalized martingale problem on D (see [Pin95, Section1.13]) is recurrent, then ζc(D) = −P and |CL−ζc(D)(D)| = 1.

Remark 2.22. The quantity ζc(D) of (2.27) is commonly referred to as the criticaleigenvalue of the operator L on D.

The structure of the eigenspaces CL−ζ(D) can differ widely depending on thechoice of the dimension k, the restrictions on the operator L, and the domain D.The case k = 1 corresponds to having a single factor and leads to eigenspaces ofdimension at most 2.

Proposition 2.23. Suppose L satisfies Assumption 2.10 on a domain D ⊆ R.Then, the number of extreme points of the convex set CL−ζ(D) is 2 for all ζ > ζc(D)and belongs to 1, 2 for ζ = ζc(D).

Remark 2.24. Proposition 2.23 reveals that, in the setting of Theorem 2.17 withk = 1, one can determine the pairing (Ψ, ν) via a three-step procedure: first, onerecovers ν by a one-dimensional Laplace inversion of u(·, y0); second, one findsΨ(ζ, y1) ν(dζ) by a one-dimensional Laplace inversion of u(·, y1) for an arbitraryy1 ∈ D\y0; third, for all ζ ≥ ζc(D) in the support of ν, one solves the second-order linear ordinary differential equation for Ψ(ζ, ·) with the obtained boundaryconditions at y0 and y1 to end up with the selection Ψ.

11


When k ≥ 2, the variability in the dimensionality of the eigenspaces is illustratedby the following two scenarios, in which the eigenspaces have dimensions 1 and ∞,respectively.

Definition 2.25. A potential P (·) on Rk, k ≥ 2, is called principally radiallysymmetric if

P = P0 + P1, (2.28)

where the functions P0 and P1 are locally integrable to power d for some d > k/2,with P0 being radially symmetric (P0(y) = P0(|y|) for some P0), and P1 vanishingoutside of a compact set.

Proposition 2.26. Let k ≥ 2 and consider a positive φ ∈ C2,η(Rk) with bounded∇φφ and ∆φ

φ , as well as an operator L := ∆ + P (y) on Rk with a locally η-Holder

continuous bounded principally symmetric potential P (·). Then, L := 1φ Lφ has the

property |CL−ζ(Rk)| = 1 for any ζ ≥ ζc(Rk) such that

ˆ ∞1

tk−3g0(t)2

( ˆ ∞t

s1−kg0(s)−2 ds

)dt =∞, (2.29)

where g0 is the unique solution of

g′′0(r) +k − 1

rg′0(r)−

(ζ − P0(r)

)g0(r) = 0 on (0,∞), g0(r) = 1 + o(r) as r ↓ 0.

(2.30)

In the situation of Proposition 2.26, we must pick Ψ(ζ, ·) as the unique elementof CL−ζ(Rk). On the other hand, in the case of a multidimensional factor processon a bounded domain D with a Lipschitz boundary, the eigenspaces are infinite-dimensional.

Proposition 2.27. Let D ⊂ Rk, k ≥ 2 be a bounded domain with a Lipschitzboundary and the coefficients a(·), b(·), P (·) of L obey (i)-(iii) in Assumption 2.10 onD. Then, the convex CL−ζ(D) has infinitely many extreme points for all ζ > ζc(D).

Thus, one cannot assert that the number of extreme points of CL−ζ(D) is finitein general. Therefore, the procedure of Remark 2.18 cannot always be reduced to afinite number of one-dimensional Laplace inversions. In such cases, we propose todetermine the FPP on a finite number of grid points y ∈ D. First, one computesthe Borel measure ν by applying the inverse Laplace transform to the left-hand sideof (2.25). Next, for each y on the grid, one calculates the selection of eigenfunctionsΨ(·, y) by taking the inverse Laplace transform of the left-hand side in (2.24). Fromhere, one can identify the value of the FPP on the grid by plugging the obtainedvalues into the equation (2.19).

3 Proof of Theorem 2.14 and a new Widder’s theorem

The goal of this section is to prove Theorem 2.14. Recall that we are interestedin separable power factor form local FPPs as in Definition 2.9. We start by focusing

12


on the function V in equation (2.7), and give a sufficient condition for V (t, x, Yt) tobe a local FPP.

Proposition 3.1. Under Assumption 2.2, let V : [0,∞)× (0,∞)×D → R be con-tinuously differentiable in t (its first argument) and twice continuously differentiablein x and y (the second and third arguments). Suppose further that V is strictlyconcave and increasing in x and a classical solution of the HJB equation

∂tV + LyV −1

2

|λ∂xV + ρκ ∂x∇yV |2

∂xxV= 0 on [0,∞)× (0,∞)×D, (3.1)

where Ly is the generator of the factor process Y . Then, V (t, x, Yt) is a local FPP.Moreover, the corresponding optimal portfolio allocations π∗ among the n stocks areof a feedback form and characterized by

σ(Yt)π∗t = −λ(Yt) ∂xV (t,Xπ∗

t , Yt) + ρκ(Yt) ∂x∇yV (t,Xπ∗t , Yt)

Xπ∗t ∂xxV (t,Xπ∗

t , Yt). (3.2)

Proof. For the former statement, one only needs to repeat the derivation of [SSZ16,equation (1.6)] mutatis mutandis and to use σ(·)σ(·)−1ρ = ρ (see Remark 2.3).For the latter statement, we apply Ito’s formula to V (t,Xπ

t , Yt) and substitute12|λ ∂xV+ρκ ∂x∇yV |2

∂xxVfor ∂tV +LyV to conclude that the drift coefficient of V (t,Xπ

t , Yt)

is 12∂xxV (t,Xπ

t , Yt) multiplied by:∣∣∣∣λ(Yt)∂xV (t,Xπt , Yt)+ρκ(Yt)∂x∇yV (t,Xπ

t , Yt)

∂xxV (t,Xπt , Yt)

+Xπt σ(Yt)πt

∣∣∣∣2. (3.3)

The process V (t,Xπt , Yt) is a local martingale if and only if the expression in (3.3)

vanishes, which happens if and only if (3.2) holds.

Remark 3.2. The process V (t, x, Yt) of Proposition 3.1 is a true FPP if V (t,Xπt , Yt)

is a true supermartingale for every π ∈ A and a true martingale for every optimalportfolio allocation π∗ of (3.2). In view of Fatou’s lemma, the supermartingale prop-erty is fulfilled if infs∈[0,t] V (s,Xπ

s , Ys) is integrable for all t ≥ 0 and π ∈ A. The mar-

tingale property is valid if the diffusion coefficients ∂xV (t,Xπ∗t , Yt)X

π∗t (σ(Yt)π

∗t )T ,

∇yV (t,Xπ∗t , Yt)κ(Yt)

T of V (t,Xπ∗t , Yt) are dt×dP-square integrable on each [0, t]×Ω.

The HJB equation (3.1) is a fully non-linear PDE and one does not expect to findexplicit formulas for its solutions in general. However, in EVE correlation marketmodels, for initial conditions of separable power type, the HJB equation (3.1) canbe linearized.

Proposition 3.3. Let ρ be an EVE correlation matrix as in Definition 2.4, and letΓ = 1−γ

γ , and q = 11+Γp . Then, the HJB equation (3.1) with an initial condition

V (0, x, y) = γγ x1−γ

1−γ h(y)q, where h > 0, has a classical solution in separable power

form, V (t, x, y) = γγ x1−γ

1−γ g(t, y), with g > 0 if and only if there exists a positivesolution to the linear PDE problem

∂tu+ Lu = 0 on [0,∞)×D, u(0, ·) = h (3.4)

13


posed in the “wrong” time direction. Hereby, L is the linear elliptic operator ofthe second order with the coefficients of (2.16). In that case, the two solutions arerelated through

V (t, x, y) = γγx1−γ

1− γu(t, y)q. (3.5)

Proof. Since we are looking for solutions of the HJB equation (3.1) in separable

power form, we plug in the ansatz V (t, x, y) = γγ x1−γ

1−γ g(t, y) to arrive at

∂tg + Lyg +Γ

2λTλg + ΓλTρκ∇yg + Γ

(∇yg)κTρTρκ∇yg2g

= 0, g(0, ·) = hq. (3.6)

Next, we employ the distortion transformation g(t, y) = u(t, y)q and get the PDE

quq−1∂tu+1

2

k∑i,j=1

(κTκ)ij(quq−1∂yiyju+ q(q − 1)uq−2(∂yiu)(∂yju)

)+

Γ

2q2uq−2(∇yu)TκTρTρκ∇yu+ q

(α+ ΓκTρTλ

)Tuq−1∇yu+

Γ

2λTλuq = 0,

(3.7)

equipped with the initial condition u(0, ·) = h. Moreover, the assumed positivity ofg translates to u > 0, so that we can divide both sides of (3.7) by uq−1. In addition,we insert the identity ρTρ = pIdB of Definition 2.4 to end up with

∂tu+1

2

k∑i,j=1

(κTκ)ij∂yiyju+(α+ ΓκTρTλ

)T∇yu+Γ

2qλTλu

+1

2u(q + Γpq − 1)(∇yu)TκTκ∇yu = 0.

(3.8)

The crucial observation is now that the non-linear term in the PDE (3.8) drops outthanks to q = 1

1+Γp . Hence, u is a positive solution of (3.4). The converse followsby carrying out the transformations we have used in the reverse order.

Proposition 3.3 reduces the task of finding solutions of the HJB equation (3.1)in separable power form to solving the linear PDE problem (3.4) set in the “wrong”time direction. The latter has been studied in [Wid63] with L being the Laplaceoperator on Rk and in [NT15] for more general linear second-order elliptic operatorson Rk. We establish subsequently a variant of [NT15, Theorem 3.12] that allows forlinear second-order elliptic operators on domains D ⊆ Rk.

Theorem 3.4. Under Assumption 2.10, a function u : (0, y0) ∪ ((0,∞)×D) →(0,∞) is a classical solution of ∂tu+Lu = 0 with u(0, y0) = 1 if and only if it admitsthe representation

u(t, y) =

ˆSL(D)

e−tζ Ψ(ζ, y) ν(dζ), (3.9)

where ν is a Borel probability measure on SL(D) and Ψ : SL(D) × D → (0,∞) isa ν-Bochner integrable selection of positive eigenfunctions. In this case, the pairing(Ψ, ν) is uniquely determined by the function u.

14


Proof. Let u : (0, y0) ∪ ((0,∞) × D) → (0,∞) be a classical solution of ∂tu +Lu = 0 with u(0, y0) = 1. Recalling the C3-diffeomorphism Ξ : D → Rk fromAssumption 2.10 and taking without loss of generality Ξ(y0) = 0 (otherwise wecompose Ξ with the translation by −Ξ(y0)) we define u : (0, 0) ∪ ((0,∞) ×Rk) → (0,∞), (t, z) 7→ u(t,Ξ−1(z)). Then, ∂tu(t, y) = ∂tu(t,Ξ(y)), ∂yiu(t, y) =∑k

j=1 ∂zju(t,Ξ(y)) ∂yiΞj(y), and

∂yiyju(t, y) =k∑

i′,j′=1

∂zi′zj′u(t,Ξ(y)) ∂yiΞi′(y) ∂yjΞj′(y)+k∑

i′=1

∂zi′u(t,Ξ(y)) ∂yiyjΞi′(y).

Plugging these into the PDE for u we conclude that u is a classical solution of∂tu+Lu = 0 with u(0, 0) = 1, where L is the operator of (2.12), (2.11) on Rk. From[NT15, Theorem 3.12] we infer that

u(t, z) =

ˆSL(Rk)

e−tζ Ψ(ζ, z) ν(dζ), (3.10)

with a Borel probability measure ν on SL(Rk) and a ν-Bochner integrable selectionof positive eigenfunctions Ψ : SL(Rk) × Rk → (0,∞) for the operator L (note thatΨ(ζ, ·) ∈ C2,η(Rk) by the Schauder interior estimate, e.g., [GT77, inequality (6.23)]).

Next, we express LΨ(ζ, ·) as

1

2

k∑i,j=1

((∇Ξi)

Ta∇Ξj)(Ξ−1(·)) ∂zizjΨ(ζ, ·)

+

k∑i=1

(((∇Ξi)

T b)(Ξ−1(·))+

1

2trace

(Hess(Ξi)a

)(Ξ−1(·))

)∂ziΨ(ζ, ·)+P (Ξ−1(·)) Ψ(ζ, ·)

= LΨ(ζ,Ξ(y))∣∣y=Ξ−1(·)

and see that LΨ(ζ, ·) = ζΨ(ζ, ·) is equivalent to LΨ(ζ,Ξ(·)) = ζΨ(ζ,Ξ(·)). Thus,SL(Rk) = SL(D) and

u(t, y) = u(t,Ξ(y)) =

ˆSL(D)

e−tζ Ψ(ζ,Ξ(y)) ν(dζ), (3.11)

where Ψ(·, ·) := Ψ(·,Ξ(·)) : SL(D)×D → (0,∞) is a ν-Bochner integrable selectionof positive eigenfunctions for the operator L (observe that the images of compactsets under Ξ are compact).

Conversely, for a function u given by (3.9), it holds u(0, y0) = 1. Moreover,defining the function u as before we find that

u(t, z) =

ˆSL(D)

e−tζ Ψ(ζ,Ξ−1(z)) ν(dζ). (3.12)

As above, we have that SL(D) = SL(Rk) and that Ψ(·, ·) := Ψ(·,Ξ−1(·)) : SL(Rk)×Rk → (0,∞) provides a ν-Bochner integrable selection of positive eigenfunctions for

15


the operator L. By [NT15, Theorem 3.12], u is a classical solution of ∂tu+ Lu = 0with u(0, 0) = 1 (here, we again assume without loss of generality that Ξ(y0) = 0).Since (∂tu+ Lu)(t,Ξ(y)) = (∂tu+ Lu)(t, y), the function u is a classical solution of∂tu + Lu = 0. Lastly, according to [NT15, Theorem 3.12] the pairing (Ψ(·, ·), ν) =(Ψ(·,Ξ−1(·)), ν

)is uniquely determined by the function u(·, ·) = u(·,Ξ−1(·)), so the

pairing (Ψ, ν) is uniquely determined by the function u.

We now have all the ingredients needed to prove Theorem 2.14.

Proof of Theorem 2.14. Take a function h : D → (0,∞), and consider


1− γg(t, y), where g > 0 and g(0, y) = h(y)q.

First, we will show that V (t, x, Yt) is a separable power factor form local FPP if andonly if V (t, x, y) is a classical solution to the HJB equation (3.1). Sufficiency followstrivially from Proposition 3.1. To prove necessity, consider a portfolio allocation

π ∈ A. We apply Ito’s formula to γγ(Xπ

t )1−γ

1−γ g(t, Yt) and infer from the conditions(ii) and (iii) in Definition 2.8 that the resulting drift coefficient must be non-positivefor all π ∈ A and equal to 0 for any maximizer π∗ ∈ A. Equating the maximumof the drift coefficient over all π ∈ A to 0 we end up with the PDE in (3.6) for g.Thus, V is a classical solution to the HJB equation (3.1).

It follows by Proposition 3.3 that V (t, x, Yt) is a separable power factor formlocal FPP if and only if g(t, y) = u(t, y)q, where

∂tu+ Lu = 0 on [0,∞)×D with u(0, ·) = h(·). (3.13)

By Theorem 3.4 each solution u of (3.13) is uniquely identified with a pairing (Ψ, ν),and is given by the right-hand side of (3.9). This yields the necessity and sufficiencyof the representation (2.18), as well as the identity (2.19). Finally, the characteriza-tion (2.20) of the optimal portfolios is a direct consequence of (3.2) and (2.19).

4 Proof of Theorem 2.17 and further ramifications

4.1 Proof of Theorem 2.17

We start our analysis of the pairing (Ψ, ν) by establishing Theorem 2.17.

Proof of Theorem 2.17. Let D′ ⊂ D be a bounded subdomain with a C3 boundary∂D′ ⊂ D and ψ : D′ → [0, 1] be a thrice continuously differentiable function withcompact support in D′. Then, Assumption 2.10 and the formulas

aij(y) =((∇Ξ−1

i )T a∇Ξ−1j

)(Ξ(y)),

bi(y) =((∇Ξ−1

i )T b)(Ξ(y)) +

1

2trace

(Hess(Ξ−1

i ) a)(Ξ(y)),

P (y) = P (Ξ(y))

(4.1)

16


render [LSU68, Chapter IV, Theorem 5.2] applicable to the problem

∂tuD′ + LuD′ = 0 pointwise on [−ε, 0]×D′, uD′ |[−ε,0]×∂D′ = 0, uD′(0, ·) = hψ(4.2)

(posed in the “right” time direction), yielding a unique classical solution with η-Holder continuous ∂tuD′ , ∂yiyjuD′ in the y variable, η

2 -Holder continuous ∂tuD′ ,

∂yiyjuD′ in the t variable, and 1+η2 -Holder continuous ∂yiuD′ in the t variable. In

particular, uD′ obeys the Feynman-Kac formula

uD′(−t, y) = E[e´ t0 P (Zs) ds (hψ)(Zt) 1τD′>t

∣∣∣Z0 = y], (t, y) ∈ [0, ε]×D′, (4.3)

where τD′ is the first exit time of Z from D′.

Using the described construction for a sequence of subdomains D′ and functionsψ increasing to D and 1D, respectively, we arrive at the monotone limit

u(−t, y) = E[e´ t0 P (Zs) ds h(Zt) 1τD>t

∣∣∣Z0 = y], (t, y) ∈ [0, ε]×D (4.4)

of uD′ , which is locally bounded on [0, ε]×D by assumption. Thanks to this and thelocal regularity estimate [LSU68, Chapter IV, inequality (10.5)] on every fixed set(−ε, 0)×D′ (and, hence, on its closure [−ε, 0]×D′) we can extract a subsequence ofuD′ converging uniformly together with ∂tuD′ , ∂yiuD′ , and ∂yiyjuD′ on every fixed

set [−ε, 0]×D′. Thus, u is a positive classical solution of the problem (2.23).

Now, consider an arbitrary positive classical solution u of the problem (2.23),and suppose that there exist pairings (Ψ(1), ν(1)) and (Ψ(2), ν(2)) such that for ally ∈ D:

h(y) =

ˆSL(D)

Ψ(1)(ζ, y) ν(1)(dζ) =

ˆSL(D)

Ψ(2)(ζ, y) ν(2)(dζ).

In view of [Pin95, Chapter 4, Theorem 3.2 and Exercise 4.16] (see also Section 4.2for more details), the elements of SL(D) are bounded below, so that the functionsu(i)(t, y) :=

´SL(D) e

−ζt Ψ(i)(ζ, y) ν(i)(dζ), i = 1, 2 are finite on [0,∞) × D. By

Theorem 3.4, each u(i) is a classical solution of

∂tu(i) + Lu(i) = 0 on (0, y0) ∪ ((0,∞)×D). (4.5)

Moreover, each

v(i)(t, y) :=

u(t, y) for (t, y) ∈ [−ε, 0]×D,u(i)(t, y) for (t, y) ∈ (0,∞)×D

(4.6)

is a positive classical solution of the PDE ∂tv(i) +Lv(i) = 0 on [−ε,∞)×D. Indeed,

on the sets [−ε, 0]×D and (0,∞)×D this PDE holds by construction, whereas

∂tu(i)(0, y) = lim

t↓0∂tu

(i)(t, y) = − limt↓0Lu(i)(t, y) = −Lu(i)(0, y), y ∈ D (4.7)

by the interior Schauder estimate of [NT15, Theorem 6.2].

17


Shifting the time by ε and renormalizing v(i), i = 1, 2 we get v(i)(t, y) :=v(i)(t−ε,y)

v(i)(−ε,y0), i = 1, 2, which solve the PDE (4.5) on [0,∞)×D. By Theorem 3.4, there

exist pairings (Ψ(i), ν(i)), i = 1, 2 such that v(i)(t, y) =´SL(D) e

−ζt Ψ(i)(ζ, y) ν(i)(dζ),

i = 1, 2. In particular, for (t, y) ∈ (0,∞)×D and i = 1, 2,

ˆSL(D)

e−ζ(t+ε) Ψ(i)(ζ, y) ν(i)(dζ) = v(i)(t+ ε, y) =v(i)(t, y)

v(i)(−ε, y0)

=

´SL(D) e

−ζt Ψ(i)(ζ, y) ν(i)(dζ)

v(i)(−ε, y0).

(4.8)

Plugging in first y = y0, then y ∈ D\y0, and relying on the uniqueness of the

Laplace transform ([Bil12, Section 30]) we read off ν(i)(dζ) = eζε

v(i)(−ε,y0)ν(i)(dζ) and

Ψ(i) = Ψ(i), i = 1, 2, from (4.8). Hence, for (t, y) ∈ (−ε, 0]×D and i = 1, 2,

u(t, y) = v(i)(t, y) = v(i)(−ε, y0) v(i)(t+ ε, y)

= v(i)(−ε, y0)

ˆSL(D)

e−ζ(t+ε) Ψ(i)(ζ, y) ν(i)(dζ)

=

ˆSL(D)

e−ζt Ψ(i)(ζ, y) ν(i)(dζ).

(4.9)

In particular, we get from the latter equation:

ˆSL(D)

e−ζt Ψ(1)(ζ, y) ν(1)(dζ) =

ˆSL(D)

e−ζt Ψ(2)(ζ, y) ν(2)(dζ).

Just like above, plugging in y = y0, then y ∈ D\y0, and utilizing the uniqueness ofthe Laplace transform we obtain ν(1)(dζ) = ν(2)(dζ) =: ν(dζ) and Ψ(1) = Ψ(2) =: Ψ.Combining this with equation (4.9) we get that any positive classical solution to theproblem (2.23) must be as given in (2.24). This yields uniqueness as desired, and inthe special case of y = y0, we obtain (2.25).

4.2 Preliminaries on positive eigenfunctions

As a preparation for the proofs of Propositions 2.21, 2.23, 2.26 and 2.27, werecall some facts about the sets SL(D) and CL−ζ(D), ζ ∈ SL(D) from the positiveharmonic function theory. Throughout the subsection we let L satisfy Assumption2.10 and infer from (4.1) that L is then locally uniformly elliptic with locally boundedand locally η-Holder continuous coefficients.

Definition 4.1 (Green’s measure). Consider the solution Z of the generalized mar-tingale problem on D associated with L0 = L − P (y) (see [Pin95, Section 1.13]).If

D′ 7→ E[ˆ ∞

0e´ t0 P (Zs) ds 1D′(Zt) dt

∣∣∣∣Z0 = y

]<∞ (4.10)

18


for all bounded subdomains D′ ⊂ D with D′ ⊂ D and y ∈ D, then the positive Borelmeasure defined by (4.10) is called the Green’s measure for L on D. The densityG(y, z) of the Green’s measure, if it exists, is referred to as the Green’s function.

By [Pin95, Chapter 4, Theorem 3.1 and Exercise 4.16] for the operators L − ζ,ζ ∈ R, we have the next proposition.

Proposition 4.2. If ζ ∈ R is such that the Green’s function exists for L − ζ, thenCL−ζ(D) 6= ∅.

We proceed to the corresponding classification of the operators L − ζ, ζ ∈ R.

Definition 4.3. An operator L − ζ on D is described as

(i) subcritical if it possesses a Green’s function,

(ii) critical if it is not subcritical, but CL−ζ(D) 6= ∅,

(iii) and supercritical if it is neither critical nor subcritical.

Thus, we are interested in the values of ζ for which L−ζ is subcritical or critical,that is, ζ ∈ SL(D). As it turns out, SL(D) is a half-line under Assumption 2.10.

Proposition 4.4 ([Pin95], Chapter 4, Theorem 3.2 and Exercise 4.16). There existsa critical eigenvalue ζc = ζc(D) ∈ R such that L − ζ is subcritical for ζ > ζc,supercritical for ζ < ζc, and either critical or subcritical for ζ = ζc.

When the potential P is non-positive, more information about the classificationof the operator L is available.

Proposition 4.5 ([Pin95], Chapter 4, Theorem 3.3 and Exercise 4.16). For anoperator L with P ≤ 0 one of the following holds:

(i) P ≤ 0, P 6≡ 0, and L is subcritical,

(ii) P ≡ 0, the solution of the generalized martingale problem on D associated withL is transient, and L is subcritical,

(iii) P ≡ 0, the solution of the generalized martingale problem on D associated withL is recurrent, and L is critical.

Remark 4.6. When γ > 1, the potential term in (2.16) is non-positive. This, puttogether with Proposition 4.5, yields 0 ∈ SL. Thus, [0,∞) ⊂ SL by Proposition 4.4.

4.3 Proofs of Propositions 2.21, 2.23, 2.26 and 2.27

At this point, we can read off Propositions 2.21, 2.23 and 2.26 from appropriateresults in [Mur86] and [Pin95].

19


Proof of Proposition 2.21. By Propositions 4.2 and 4.4,

infζ ∈ R : ζ ∈ SL(D)

= ζc(D) ∈ SL(D). (4.11)

If P is constant and the solution of the generalized martingale problem onD for L−Pis recurrent, then L−P is critical by Proposition 4.5, and hence, ζc(D) = −P . In thiscase, [Pin95, Chapter 4, Theorem 3.4 and Exercise 4.16] yield |CL−ζc(D)(D)| = 1.

Proof of Proposition 2.23. It suffices to put together Proposition 4.4 with [Pin95,Chapter 4, Remark 2 on p. 149, Theorem 3.4 and Exercise 4.16].

Proof of Proposition 2.26. Note that, for any ζ ≥ ζc(Rk) and f ∈ CL−ζ , one hasφf ∈ CL−ζ . Therefore, it is enough to prove |CL−ζ | = 1, ζ ≥ ζc(Rk), which is

readily obtained by combining Proposition 4.4 with [Mur86, Theorem 5.3].

Remark 4.7. The condition (2.29), on L and ζ, needs to be verified on a case-by-case basis. For example, consider a locally η-Holder continuous non-positive boundedradially symmetric potential P0 with P0(r) = cr−2, r ≥ 1 for some c < 0. TakeL = 1

φ(∆ + P0)φ for some φ as in Proposition 2.26 and ζ = 0. Then,

g0(r) = c1r2−k+

√(k−2)2−4c2 + c2r

2−k−√

(k−2)2−4c2 , r ≥ 1 (4.12)

for some c1, c2 ∈ R. By [Mur86, Theorem 4.6(iii) and Theorem 2.4(ii)] the operator∆+P0 is subcritical, so that c1 6= 0 by [Mur86, Theorem 3.1(ii)]. Thus, (2.29) holds.

In the context of Proposition 2.27, the structure of the sets CL−ζ(D), ζ > ζc(D)has been described in [Anc78, Theorems 6.1 and 6.3], which we briefly recall for theconvenience of the reader.

Definition 4.8 (Minimal eigenfunction). A function f ∈ CL−ζ(D) is referred to as

minimal if f ≤ f implies f = f for all f ∈ CL−ζ(D).

Proposition 4.9 ([Anc78], Theorems 6.1 and 6.3). In the setting of Proposition2.27, every minimal element f ∈ CL−ζ(D) has the property limz→y f(z) > 0 forexactly one point y ∈ ∂D and is uniquely determined by y. In addition, for everyf ∈ CL−ζ(D), there exists a unique Borel probability measure ξ on ∂D such that

f(·) =

ˆ∂D

fy(·) ξ(dy), (4.13)

where fy is the minimal eigenfunction associated with y.

Proposition 2.27 is a direct consequence of Proposition 4.9.

Proof of Proposition 2.27. The uniqueness of the Borel probability measure ξ in therepresentation (4.13) shows that the extreme points of CL−ζ(D) are precisely theminimal eigenfunctions fy, y ∈ ∂D. Clearly, |fy : y ∈ ∂D| = |∂D| =∞.

20


5 Merton problem in stochastic factor models

In this section, we consider the framework of the Merton problem, in which aninvestor aims to maximize her expected terminal utility from the wealth acquiredthrough investment:

supπ∈A

E[υT (XπT , YT )]. (5.1)

Thereby, the time horizon T and the utility function υT are chosen once and forall at time zero. It is well-known (see e.g. [FS06, Section IV.3]) that the dynamicprogramming equation for the Merton problem within the Markovian diffusion model(2.1)-(2.3) takes the shape of the HJB equation

∂tV + LyV −1

2

|λ∂xV + ρκ ∂x∇yV |2

∂xxV= 0. (5.2)

In contrast to the preceding discussion, here the HJB equation is equipped witha terminal condition V (T, ·, ·) = υT and, hence, posed in the backward (“right”)time direction. It turns out that, under Definition 2.4, we can reduce the backwardproblem to a linear second-order parabolic PDE posed in the “right” time direction,provided that the terminal utility function is of separable power form: υT (x, y) =

γγ x1−γ

1−γ gT (y), and that appropriate technical assumptions hold.

Theorem 5.1. Let γ ∈ (0, 1). Suppose the market model (2.1)-(2.3), the correlationmatrix ρ, and the linear elliptic operator of the second order L with the coefficients

a(·) = κ(·)Tκ(·), b(·) = α(·) + Γκ(·)TρTλ(·), P (·) =Γ

2qλ(·)Tλ(·) (5.3)

satisfy Assumptions 2.2, 2.4, and 2.10, respectively, where Γ = 1−γγ and q = 1

1+Γp .Suppose further that the volatility matrix κ(·) of the factor process is bounded, theweak solution Z of the SDE associated with L0 = L − P (y) remains in D, and the

terminal utility function is of separable power form υT (x, y) = γγ x1−γ

1−γ h(y)q, with an

h ∈ C2,η(D) bounded above and below by positive constants and such that

(t, y) 7→ ∇y E[e´ t0 P (Zs) ds h(Zt)

∣∣Z0 = y]

(5.4)

is bounded on [0, T ] × D. Then, the value function for the corresponding Mertonproblem, V (t, x, y) = supπ∈A E[υT (Xπ

T , YT ) |Xπt = x, Yt = y], can be written as


1− γu(t, y)q. (5.5)

Hereby, u is a classical solution of the linear PDE problem

∂tu+ Lu = 0 on [0, T ]×D, u(T, ·) = h. (5.6)

Moreover, every portfolio allocation π∗ fulfilling

σ(Yt)π∗t =

1

γ

(λ(Yt) + qρκ(Yt)

∇yu(t, Yt)

u(t, Yt)

)(5.7)

is optimal.

21


Proof. By the classical verification paradigm (see e.g. [FS06, Chapter IV, proof ofTheorem 3.1]), it is enough to show that for every portfolio allocation π ∈ A theprocess V (t,Xπ

t , Yt), t ∈ [0, T ] is a supermartingale, and that for every solution π∗

of (5.7) the process V (t,Xπ∗t , Yt), t ∈ [0, T ] is a martingale.

We follow the proof of Proposition 3.3 in the reverse direction and find thatg(t, y) := u(t, y)q is a classical solution of the problem (3.6), whereas the function Vdefined by (5.5) is a classical solution of the HJB equation (5.2) with V (T, ·, ·) = υT .For any π ∈ A, we may now apply Ito’s formula to V (t,Xπ

t , Yt) and replace ∂tV+LyVby 1

2|λ ∂xV+ρκ ∂x∇yV |2

∂xxVto see that the drift coefficient of V (t,Xπ

t , Yt) is the product

of 12∂xxV (t,Xπ

t , Yt) with the expression in (3.3) and, in particular, non-positive.Hence, the local martingale part of V (t,Xπ

t , Yt) is bounded below by −V (0, x, y)and, consequently, a supermartingale. Thus, V (t,Xπ

t , Yt) is a supermartingale aswell.

Next, we deduce from the proof of Theorem 2.17 that u(t, y) admits the stochasticrepresentation

u(t, y) = E[e´ T−t0 P (Zs) ds h(ZT−t)

∣∣∣Z0 = y]

(5.8)

(recall that Z remains in D by assumption). In addition, our further assumptionsimply that ∇yu is bounded on [0, T ] ×D, and that u is bounded above and belowby positive constants on [0, T ]×D. Together with the boundedness of the volatilitymatrix κ(·) of the factor process and the Sharpe ratio λ(·) (see Assumption 2.10(i))this yields the boundedness of σ(Yt)π

∗t via (5.7). Finally, the drift coefficient of

V (t,Xπ∗t , Yt) vanishes and the quadratic variation of its local martingale part com-

putes to

ˆ t

0γ2γ (Xπ∗

s )2−2γ |σ(Ys)π∗s |2 +

γ2γq2

(1− γ)2(Xπ∗

s )2−2γ u(s, Ys)2q−2 |κ(Ys)∇yu(s, Ys)|2

+2γ2γq

1− γ(Xπ∗

s )2−2γ u(s, Ys)q−1 (σ(Ys)π

∗s)Tρκ(Ys)∇yu(s, Ys) ds.

(5.9)

The expectation of the latter integral is finite for all t ∈ [0, T ], since σ(Ys)π∗s and

u(s, Ys)q−1κ(Ys)∇yu(s, Ys) are bounded, while supt∈[0,T ] E[(Xπ∗

t )2−2γ ] < ∞ thanksto the boundedness of σ(Ys)π

∗s and λ(Ys) in

Xπ∗t = x exp

( ˆ t

0(σ(Ys)π

∗s)Tλ(Ys) ds+

ˆ t

0(σ(Ys)π

∗s)T dWs −

1

2

ˆ t

0|σ(Ys)π

∗s |2 ds

).

(5.10)We conclude that V (t,Xπ∗

t , Yt) is a true martingale.

6 Discussion of EVE assumption

This last section is devoted to a thorough investigation of Definition 2.4 thatplays a key role in the proof of Theorem 2.14. It is instructive to start with the twoextreme cases corresponding to taking p = 1 and p = 0 therein, respectively. Suppose

22


first that A = 0 in (2.3), in other words, the components of the Brownian motionB driving the factors are given by linear combinations of the components of theBrownian motion W driving the stock prices. We can then reparametrize the modelsuch that B = W , ρ = IdW , and ρTρ = IdW . Consequently, Definition 2.4 holds withp = 1. The resulting market is complete, and we find ourselves in the framework of[NT15, Section 2.3]. It is therefore not surprising that the HJB equation (3.1) canbe reduced to a linear PDE, even though the linearization in Proposition 3.3 differsfrom the one in [NT15, Section 2.3]. On the other hand, when ρ = 0 in (2.3), theBrownian motions B and W become independent, leading to an incomplete market.Nonetheless, Definition 2.4 is still satisfied with p = 0. Thus, the linearization inProposition 3.3 goes far beyond the complete market setup.

More generally, Definition 2.4 can be put to use as follows. In practice, thecorrelation matrix ρ can have hundreds or thousands of entries and, hence, might bedifficult to estimate accurately in its entirety. However, one can attempt to obtain aless noisy estimate by projecting an estimate for ρ onto the submanifold of dW × dBmatrices fulfilling Definition 2.4. Restricting the attention to the non-trivial casedW ≥ dB (see Remark 2.6), with the exception of the zero matrix, the latter matricescan be written uniquely as rQ, where r ∈ (0, 1] and Q has orthonormal columns,

thereby forming a(1 + dWdB − dB(dB+1)

2

)-dimensional submanifold of RdW×dB . As

it turns out, the most tractable projection onto this submanifold is that with respectto the Frobenius norm (also known as the Hilbert-Schmidt norm) on RdW×dB .

6.1 Choice of r and Q

Let us equip the space RdW×dB with the Frobenius norm:

|A|F =

( dW∑i=1

dB∑j=1

a2ij

)1/2

=(traceATA

)1/2. (6.1)

For an estimate ρ of ρ, we are able to find a constant r ∈ [0, 1] and a matrix withorthonormal columns Q that minimize the distance induced by the Frobenius norm.

Proposition 6.1. Consider the minimization problem

min |ρ− rQ|F such that r ∈ [0, 1], QTQ = IdB . (6.2)

Then, r∗ = trace(ρT ρ)1/2

dBand Q∗ = ρ(ρT ρ)−1/2 are the minimizers.

Proof. Equivalently, consider the problem

min |ρ− Q|2F such that QT Q = r2IdB (6.3)

for fixed r ∈ [0, 1] and minimize over r ∈ [0, 1] subsequently. Applying the methodof Lagrange multipliers with a dB × dB Lagrange multiplier matrix Λ we get

2(Q− ρ) = Q(Λ + ΛT ) ⇐⇒ Q(2IdB − Λ− ΛT ) = 2ρ. (6.4)

23


Passing to the transpose on both sides of the last equation, taking the product ofthe resulting equation with the original equation, and recalling the constraint we see

r2(2IdB − Λ− ΛT )2 = 4ρT ρ ⇐⇒ r(2IdB − Λ− ΛT ) = 2(ρT ρ)1/2, (6.5)

where (ρT ρ)1/2 is the dB × dB square root of the matrix ρT ρ. Together with (6.4)and the notation (ρT ρ)−1/2 for the inverse of (ρT ρ)1/2 this yields

Q = rρ(ρT ρ)−1/2. (6.6)

Plugging the formula for Q back into the objective function we are left with theminimization problem

minr∈[0,1]

∣∣ρ− rρ(ρT ρ)−1/2∣∣2F⇐⇒ min

r∈[0,1]

(trace(ρT ρ)− 2r trace(ρT ρ)1/2 + r2dB

).

(6.7)

Consequently, the optimal r is trace(ρT ρ)1/2

dB, that is, the average of the singular values

of ρ, whereas Q should be picked according to (6.6).

6.2 Choice of p

If one is only interested in the parameter p from Definition 2.4, then it is mostnatural to minimize |ρT ρ− pIdB | for a selection of a norm | · | on RdB×dB . When | · |is the operator norm (also known as the spectral radius or the Ky Fan 1-norm),

|ρT ρ− pIdB | = max1≤i≤dB

|θi − p|, (6.8)

where θ1 ≤ θ2 ≤ · · · ≤ θdB are the ordered eigenvalues of ρT ρ (or, equivalently, theordered squared singular values of ρ). In this case, |ρT ρ − pIdB | is minimized by

p =θ1+θdB

2 . When | · | is the Frobenius norm,

|ρT ρ− pIdB | =( dB∑i=1

|θi − p|2)1/2

. (6.9)

The minimizer for the latter is p =θ1+θ2+···+θdB

dB. When | · | is the trace norm (also

known as the nuclear norm or the Ky Fan dB-norm),

|ρT ρ− pIdB | =dB∑i=1

|θi − p|, (6.10)

which is smallest for the median of θ1, θ2, . . . , θdB.

6.3 Example: affine factor models

We conclude by illustrating the use of the EVE assumption in the framework ofaffine market models with non-negative factors. In that situation, both the forwardinvestment problem and the Merton problem can be reduced to the solution of

24


a system of Riccati ordinary differential equations (ODEs). Consider the affinespecialization of the factor model (2.1)-(2.3):

dSitSit

= µi(Yt) dt+

dW∑j=1

σji(Yt) dW jt , i = 1, 2, . . . , n, (6.11)

dYt = (MTYt + w) dt+ κ(Yt)T dBt, (6.12)

Bt = ρTWt +ATW⊥t , (6.13)

where M has non-negative off-diagonal entries, w ∈ [0,∞)k, and µ(·), σ(·), κ(·), ρare such that

λ(y)Tλ(y) = µ(y)Tσ(y)−1(σ(y)T

)−1µ(y) = ΛT y, (6.14)

κ(y)Tκ(y) = diag(L1y1, L2y2, . . . , Lkyk) with L1, L2, . . . , Lk ≥ 0, (6.15)

Γκ(y)TρTλ(y) = NT y. (6.16)

Remark 6.2. The condition (6.15) is necessary for the process Y of (6.12) to be[0,∞)k-valued and affine (see [FM09, Theorem 3.2]). Conversely, the SDE (6.12)with volatility coefficients satisfying (6.15) has a unique weak solution, which is affineand takes values in [0,∞)k (see [FM09, Theorem 8.1]).

Suppose now that the initial utility function for the forward investment problemor the terminal utility function for the Merton problem is of separable power formwith h(y) = exp(HT y + h0). Under the EVE assumption, the HJB equation (3.1)arising in the two problems can be transformed into the linear second-order parabolicPDE of (3.4) (see the proof of Proposition 3.3), which in the setting of (6.11)-(6.16)amounts to

∂tu+1

2

k∑i=1

Liyi∂yiyiu+ yT (M +N)∇yu+ wT∇yu+Γ

2qyTΛu = 0. (6.17)

Inserting the exponential-affine ansatz u(t, y) = exp(ΦTt y + Θt) we obtain

yT Φt + Θt +1

2

k∑i=1

Liyi(Φit)

2 + yT (M +N)Φt + wTΦt +Γ

2qyTΛ = 0. (6.18)

Equating the linear and the constant terms in y to 0 leads to the following systemof Riccati ODEs:

Φit +

1

2Li(Φ

it)

2 +k∑j=1

(M +N)ijΦjt +

Γ

2qΛi = 0, i = 1, 2, . . . , k, (6.19)

Θt + wTΦt = 0. (6.20)

We note that Θ is completely determined by the solution Φ of the system (6.19).The latter can be solved numerically in general and, for special kinds of M and N ,

25


even explicitly. For example, when M and N are diagonal the system (6.19) splitsinto k one-dimensional Riccati ODEs:

Φit +

1

2Li(Φ

it)

2 + (Mii +Nii)Φit +

Γ

2qΛi = 0, i = 1, 2, . . . , k. (6.21)

These ODEs can be solved by a separation of variables and subsequent integration.For instance, when γ ∈ (0, 1) and the discriminants Di := (Mii + Nii)

2 − LiΓqΛi

associated with the quadratic equations 12Liz

2 +(Mii+Nii)z+ Γ2qΛi = 0 are positive

for all i, we obtain the (real) roots

z+,i =−Mii −Nii +

√Di

Li, z−,i =

−Mii −Nii −√Di

Li. (6.22)

The general solution of (6.21) then becomes

Φit =

z+,i − χi z−,i e−√Dit

1− χi e−√Dit

, i = 1, 2, . . . , k, (6.23)

and one can find the constants χi by setting Φ to H at the terminal time (for theMerton problem) or at time 0 (for the forward investment problem).

We conclude by discussing, in the latter setting and with Mii +Nii ≥ 0 for all i,the true FPP property of the process

V (t, x, Yt) = γγx1−γ

1− γu(t, Yt)

q = γγx1−γ

1− γexp

(qΦT

t Yt + qΘt

). (6.24)

By arguing as in the second paragraph of the proof of Theorem 5.1 we concludethat V (t,Xπ

t , Yt) is a true supermartingale for each π ∈ A. It remains to see ifV (t,Xπ∗

t , Yt) is a true martingale for some π∗ ∈ A as in (3.2). To this end, weconsider the expectation of the integral in (5.9). In view of the Cauchy-Schwarzinequality and Fubini’s theorem, it suffices to control the expectations of the twosummands in the first line of (5.9) uniformly over s ∈ [0, t]. The random variablesentering the two summands read in the case at hand as follows:

Xπ∗s =x exp

(ˆ s

0

(ΛT

γ+qΦT

r NT

1−γ

)Yr dr +

ˆ s

0(σ(Yr)π

∗r )T dWr −

1

2

ˆ s

0|σ(Yr)π

∗r |2 dr

),

|σ(Ys)π∗s |2 =

1

γ2

(ΛTYs +

2q

ΓΦTs N

TYs + pq2k∑i=1

Li(Φis)

2Y is

),

|κ(Ys)∇yu(s, Ys)|2 = u(s, Ys)2

k∑i=1

Li(Φis)

2Y is .

With p1, p2 > 1 satisfying p−11 + p−1

2 = 1, γ := 1 − γ and p3 := 1 − 2γp2 <1, we bound the expectation of the first summand in the first line of (5.9) using

26


Holder’s inequality and the supermartingale property of stochastic exponentials (seee.g. [KS91, Chapter 3, discussion before Proposition 5.12]) by γ2γx2γ times

E[|σ(Ys)π

∗s |2p1 exp

(2p1γ

ˆ s

0(σ(Yr)π

∗r )Tλ(Yr) dr − p1γp3

ˆ s

0|σ(Yr)π

∗r |2 dr

)] 1p1

· E[exp

(2p2γ

ˆ s

0(σ(Yr)π

∗r )T dWr − p2γ(1−p3)

ˆ s

0|σ(Yr)π

∗r |2 dr

)] 1p2

≤ E[|σ(Ys)π

∗s |2p1 exp

(2p1γ

ˆ s

0

(ΛT

γ+qΦT

r NT

1−γ

)Yr dr−p1γp3

ˆ s

0|σ(Yr)π

∗r |2 dr

)] 1p1

.

For every i and r, the coefficient of Y ir in the latter exponential admits the estimate

p1γ

(2Λiγ

+2qNiiΦ

ir

1− γ− p3

γ2

(Λi +

2q

ΓNiiΦ

ir + pq2Li(Φ

ir)

2))

≤ p1γ

(Λi

2γ − p3

γ2+Niic

Φi,1

2q(γ − p3)

γ(1− γ)− Li(cΦ

i,2)2 pq2p3

γ2

)=: βi,

(6.25)

where

cΦi,1 =

z+,i if Nii ≥ 0,

z−,i if Nii < 0and cΦ

i,2 =

z+,i if p3 ≥ 0,

z−,i if p3 < 0(6.26)

(note that (6.23) and Mii +Nii ≥ 0 imply z−,i ≤ Φis ≤ z+,i ≤ 0).

In view of the uniform boundedness of any given moment of Ys over s ∈ [0, t](see [FM09, Lemma A.1, Lemma 2.3(iv) and Theorem 3.2]) and Holder’s inequality,it suffices to control the exponential moment of

´ s0 Yr dr of an order slightly larger

(componentwise) than β := (β1, β2, . . . , βk) uniformly over s ∈ [0, t]. With theexplicit solution

−Mii +√

∆i tan

(arctan

Mii√∆i

+

√∆i

2t

), i = 1, 2, . . . , k (6.27)

to the system of Riccati ODEs in [FM09, Theorem 4.1(ii), third line of display (4.5)],where ∆i = 2Liiβi −M2

ii, we find that the exponential moment in consideration isbounded uniformly over s ∈ [0, t] as long as

t < mini=1,2,...,k

π − 2 arctan(Mii/√

∆i)√∆i

. (6.28)

Similarly, the expectation of the second summand in the first line of (5.9) is less or

equal to γ2γq2

(1−γ)2e2qΘs times

E[( k∑

i=1

Li(Φis)

2Y is

)p1exp

(2p1γ

ˆ s

0

(ΛT

γ+qΦT

r NT

1−γ

)Yr dr−p1γp3

ˆ s

0|σ(Yr)π

∗r |2 dr

)] 1p1

,

(6.29)which is also bounded uniformly over s ∈ [0, t] as long as (6.28) holds. All in all, theprocess in (6.24) is a true FPP at least until (but possibly not including) the timeon the right-hand side of (6.28).

27


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